Law of the iterated logarithm for random graphs

Early View

Abstract

A milestone in probability theory is the law of the iterated logarithm (LIL), proved by Khinchin and independently by Kolmogorov in the 1920s, which asserts that for iid random variables
{ti}i=1∞ with mean 0 and variance 1

In this paper we prove that LIL holds for various functionals of random graphs and hypergraphs models. We first prove LIL
for the number of copies of a fixed subgraph H. Two harder results concern the number of global objects: perfect matchings and Hamiltonian cycles. The main new ingredient
in these results is a large deviation bound, which may be of independent interest. For random k‐uniform hypergraphs, we obtain the Central Limit Theorem and LIL for the number of Hamilton cycles.

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